# How did egyptian mathematical fractions evolve in antiquity?

There are certain explanations on how integers might have evolved, like for example "the wings of a bird to symbolize the number two, clover--leaves three, the legs of an animal four, the fingers on his own hand five."$_1$ Seeing all these, and making experience short--abstract, can be said to have given the numbers$_2$. Egyptian integer number notation can itself be traced back to their notion of marks on tally (for the sign of one), hobble for cattle (for ten), measuring rope (for hundred).$_3$

Similarly what are the explanations given for the evolution of fractions in Egypt, by any of the found evidence? Did division discovery gave raise to fractions or fractions discovery gave raise to division, in Egypt?

$_1$Tobias Dnatzig, Number: The Language of Science
$_2$This expression can be found in Hamilton's letter of Sep.16, 1828, in the book Life of Sir William Rowan Hamilton.
$_3$Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History Page No.18-21.

• I've posted an answer to your question. The first part may not be entirely relevant, but I thought it added some nice background. I hope I have addressed both of the questions you have posed at the end of my answer. – Nick R May 13 '16 at 0:52

According to Carl Boyer, in his text A History of Mathematics, the earliest record we have regarding fractions in ancient Egypt comes from the Ahmes Papyrus, popularly known as the Rhind Papyrus, circa 1650 BCE. The scribe, Ahmes, tells us that it is derived from a prototype of the Middle Kingdom, circa 2000-1800 BCE, and it is possible that some of this knowledge may be handed down from as early as 3000 BCE.

According to Boyer, the general concept of fraction appears to have been an enigma to the Egyptians. They did not regard the general rational fraction $\frac{n}{m}$ as an elementary "thing", but as an uncompleted process.

Where today we think of $\frac{3}{5}$ as a single irreducible fraction, Egyptian scribes thought of it as reducible to sum of the three unit fractions $\frac13$ and $\frac15$ and $\frac1{15}$.

With this in mind, the papyrus begins with two tables; the first expressing fractions of the form $\frac{2}{n}$ for odd $n$ from $5$ to $101$ expressed as the sum unit factions ($\frac{1}{m}$). A second, shorter table follows giving a similar treatment for fractions of the form $\frac{n}{10}$ for $n = 1$ to $9$.

(To complicate things, in addition to unit fractions, the Egyptians also appear to have been comforatble with the fraction $\frac23$, which appears in the tables. For example $\frac9{10}$ is given as $\frac{1}{30} + \frac15 + \frac23$.)

Following the two tables is a list of eighty-four assorted problems which may provide some motivation for the evolution of fractions in Egypt. Typical of these problems is the division of a given number of loaves of bread amongst ten men.

According to Boyes :

The fundamental arithmetic operation in Egypt was addition, and our operations of multiplication and division were performed in Ahmes' day through successive doubling, or "duplication".

This suggests that the concept of fraction preceded the formal concept of division.

• +1. Thank you for the help. But, I think there is a need for greater details and evidence on the evolution of mathematics, in the way of details expressed in my question. I don't know whether there exists such data or not to this day, I am searching for it, I haven't found it. If you find anything more do share it, if I find anything I will do the same :) – Immortal Player May 13 '16 at 1:55